====== Josh's Side Project ======

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Josh described in today's lab meeting a model to explore the Red Queen hypothesis for sex recombination.  In the scenario there is a host/parasite system where the parasites "chase" host genotypes and the hosts escape.  It's thought that recombination enhances their escape rate and so it's selected for.  Josh explored a discrete-time, deterministic model but wondered if an individual-based approach would work as well.  Here's how I might tackle it:

===== Host logistics =====

I'd start with a host population that obeys logistic dynamics, with an asexual birth process and competitive death process.  It's artificial but about the simplest you can get and very well understood.  For a focal individual with heritable traits \(i\) the reactions are
\[
\begin{array}{rcl}
  H_i & \xrightarrow{\beta} & 2 H_i \\
  H_i + H_j & \xrightarrow{\delta} & H_j.
\end{array}
\]

The density of individuals with a particular trait \(h_i\) would then follow, in the limit of large numbers (from [[wp>mass action]]),
\[
  \diff{h_i}{t} = h_i (\beta - \delta h)
\]
where \(h\) is the total density marginalized over all traits, \(h=\sum_j h_j\).  In the case of a clonal population the dynamics reduce to logistic.

===== Predators =====

We want a predator that subsists on the hosts and dies in its absence.  For a Red Queen effect we need the predator's success to depend on the traits of the host and predators:
\[
\begin{array}{rcl}
  H_i + P_j & \xrightarrow{\alpha(i, j)} & 2 P_j \\
  P_j & \xrightarrow{\gamma} & \emptyset.
\end{array}
\]

The population dynamics for this system would then become
\[
\begin{array}{rcl}
  \diff{h_i}{t} & = & h_i \left( \beta - \delta h - \sum_j \alpha(i,j) p_j \right) \\
  \diff{p_j}{t} & = & p_j \left( \sum_i \alpha(i,j) h_i - \gamma\right),
\end{array}
\]
a generalized Lotka-Volterra predator-prey model.  In a monoclonal scenario it reduces to the Rosenzweig-MacArthur model sans satiation (ie. a Type-1 functional response),
\[
\begin{array}{rcl}
  \diff{h}{t} & = & h \left( \beta - \delta h - \alpha p \right) \\
  \diff{p}{t} & = & p \left( \alpha h - \gamma\right),
\end{array}
\]

===== Trait variation =====

To study the Red Queen hypothesis we need a source of trait variation.  Ideally, this should be introduced in the birth process but it can be tricky to introduce a mechanism that doesn't seriously complicate things.  So let's separate out births and trait changes.  Instead we'll have existing individuals change their traits through new processes which, if desired, can be thought of as a birth/death replacement process.

We want to study recombination in the hosts so we're free to use a simpler process in the predators--let's apply spontaneous mutation:
\[
  P_j \xrightarrow{\mu(j,k)} P_k.
\]

Now we're ready to introduce recombination of the hosts.  The simplest possibility is an interaction between a focal individual \(i\) and a partner such that the focus "inherits" some of the traits of the partner:
\[
  H_i + H_l \xrightarrow{\rho(i,l,m)} H_m + H_l
\]
where \(H_m\) represents an individual with a mix of traits \(i\) and \(l\).  Minimally, a single allele could be replaced.

===== Mean field =====

In the large, well-mixed limit the population dynamics for the system of all the above processes becomes
\[
\begin{array}{rcl}
  \diff{h_i}{t} & = & h_i \left( \beta - \delta h - \sum_j \alpha(i,j) p_j \right)  + \sum_{l,m} \left( \rho(m,l,i) h_m - \rho(i,l,m) h_i \right) h_l \\
  \diff{p_j}{t} & = & p_j \left( \sum_i \alpha(i,j) h_i - \gamma\right) + \sum_k \left( \mu(k,j) - \mu(j,k) \right) p_k.
\end{array}
\]
